Category: Math Standards

Like many of my generation, life changed a bit when Sputnik was launched in October, 1957. While many of my classmates were interested in rocketry, my own interest was more in the field of electronics – the instruments needed to make measurements of temperature, pressure, and other data that was then sent back to Earth by radio. Because I was an amateur radio operator at the time (K9SRW) who built all my own equipment, this was a natural extension of then-current interests. I remember walking part way home from high school just so I could stop at a local Army surplus store packed with boxes of resistors, capacitors, and other components including the transistors needed to build amplifiers, oscillators, and other circuits one might need.

Since, as I said, rocketry was not my goal, I looked for any way to get a project off the earth, even if it didn’t go into orbit. As a result, in 1961, the method I chose (helium-filled weather balloons) was not only inexpensive, it could be used to carry a pretty heavy payload (two kg or so). With my focus on the electronics, I built the transmitter, and the attachments needed to measure altitude, temperature, air pressure, luminosity, and to send the legally required Identification signal. All of these circuits were modular, and a lot of time was spent making sure everything worked. My father provided a photographic plate to see if I could detect cosmic rays (assuming I would get the plate back from the experiment so it could be developed.)

The finished payload was a cube about 30 cm on a side, and I built two of them – one in Styrofoam for launch, and one in clear plastic for testing and display for a science fair at my high school. I called the experiment Project HiBall (for high balloon, of course) and on launch day I just hoped everything worked.

Fortunately, the experiment was a success. The balloon headed west, and landed a day later on a farm in Iowa where a kind farmer found it and sent it to me. The data was not earth-shattering, but the experiments mostly worked as planned and the resulting science fair project was well-received, taking me to the State finals. While my interest in STEM subjects had already been formed, there is little question that this project strengthened these interests, setting the trajectory for my continued education.


The reason I shared this experience with you is because, today, even more amazing options are available. The first technology to mention is the CubeSat- small (10 cm/side, about 1 kg)) satellites for student projects that stay in low-earth orbits for about a year ( While most of the projects are done by college students, there is a special opportunity to expand this access to high school students. This project (ArduSat – is based on the popular Arduino board used to send and receive data from all kinds of sensors and actuators. While most Arduino projects reside here on Earth, the Ardusat system lets students design and test experiments in their classroom that can then be sent to an Arduino-based CubeSat for testing in space. From my historical perspective, this is staggering!ardusat

The Arduino board connects to a computer and has numerous inputs and outputs for both digital and analog data. The Ardusat student kit includes some special sensors for luminosity, temperature, an accelerometer, gyroscope, magnetometer, barometer, UV sensor, infrared thermopile and other data sources. The whole kit is only $150 which is a bargain considering the specialized sensors it contains. While experiments can be designed and tested here on Earth, finished Arduino programs can be sent 450 km up to the Ardusat where experiments can be done and the data sent to Earth.

This goes way beyond what I was doing in 1961 in two very important ways. First, the experiments are done on an orbiting satellite. Second, the projects can be done by students without them having to design all the sensors and other equipment themselves. This has the effect of democratizing the endeavor, bringing an amazing opportunity for STEM education to students everywhere.

In addition to the hardware kits, Ardusat also has a lot of activities and experiments that can be downloaded and explored – including tutorials on the hardware itself. This material is generally released under a Creative Commons copyright, making it perfect for free classroom use.

In addition to the tutorials and other resources, the activities are keyed to both the Next Generation Science and the Common Core Standards. This adds value in that teachers can see how Ardusat projects tie into the standards they are expected to support without having to wade through the massive standards documents themselves.

There is no question in my mind that the project I did ages ago helped guide me into the sciences. What excites me more is that projects like Ardusat will achieve this result for thousands of kids who well then go on to invent our future.

The Trinity Fractal

The story behind this discovery dates back to the 1970’s when I used to volunteer as a math resource specialist at a small school near my office.  One day, a teacher introduced me to a 10 year-old girl (we will call her “Ann”) who was (in the teacher’s  words) “bad at math.”  I found that to be  strange announcement since, in my view, Ann had never been exposed to math, but only to arithmetic.  So one day I invited my class to do an experiment.  

I brought my own bucket of pattern blocks to school, in which I added some regular pentagons I had made.  Piles of the same shapes were put on several desks, and students were asked to tile the surface with just one shape – and I made sure Ann was at the table with the regular pentagons.  After a few minutes, all the students had succeeded – excepting those at the table with the pentagons.  No matter how hard they tried, there was always going to be a gap.

Ann said, “Well, we can do it, but we are going to need a lot of grout.”  And, after looking at the other tables, she said, “This is a strange kind of math – 3 works, 4 works, and 6 works, but 5 doesn’t work.  Why is that?”

I was delighted to hear that question because this is the kind of question mathematicians ask themselves often.  Ann told me she wanted to experiment more with the pentagons, and I gave her a bunch to take home so she could report her findings the next week when we met again.  At this point I didn’t know what to expect, but it sure wasn’t what she showed up with.

The next week, she started off with the following pattern:

Starting pattern

Ann pointed out that this shape, while not a tiling pattern, looked like a pentagon if you “squished” your eyes a little bit.  “So,” she said, “suppose we start with this pattern, shrink it in size and build a new pattern with the same shape.”

First generation pattern

Then, she said, just keep repeating this process.  You will always need some grout, but the picture should be very pretty.

The next two generations of patterns are shown below:

Second generation

Third generation

As you can see, Ann was quite right.  Yes, you still need grout, and, the resulting pattern is quite pretty.

Basically, what Ann had discovered (and accurately described) is a fractal – a shape with a non-integer dimension.  I’ve told Ann’s story many times, but never before constructed the fractal patterns to show people.  I decided to call this the Trinity fractal, named after the school where I volunteered (Trinity Parish School in Menlo Park, California.)

I lost touch with Ann, but talked with her on her first day of college at UC Berkeley, where she was majoring in mathematics.

I’m glad I may have played a small role in helping her see the beauty in this subject, and have no doubt that she has gone on to do great things.

I’m also happy to finally share her discovery with others in the hope that it encourages other teachers to move beyond arithmetic to see the beauty in real mathematics, as encouraged by the Common Core State Standards for mathematics.  Our workshops on CCSS Math can be scheduled by e-mailing me at  Also, our work with pentagon tiling has continued in a new way.  See to see what we’re doing in this area!

I want to start with an observation about the Common Core State Standards for mathematics.  In the past, I have been openly critical of many standards, but I find the central ideas behind this set quite appealing.  The key point is that these standards point the way toward student understanding of critical mathematical ideas, not just the memorization and regurgitation of facts quickly forgotten after the test.

Here are the key concepts:
  1. Make sense of problems and persevere in solving them.
  2. Reason abstractly and quantitatively.
  3. Construct viable arguments and critique the reasoning of others.
  4. Model with mathematics.
  5. Use appropriate tools strategically.
  6. Attend to precision.
  7. Look for and make use of structure.
  8. Look for and express regularity in repeated reasoning.

These are all wonderful ideas that directly relate to how mathematicians work.  The challenge is how these can be implemented in the teaching of mathematics.  One example I find interesting is to see how we can use 3D printers in the math classroom in support of these core ideas.  For example (and I have several), consider the construction of polyhedra using plastic struts and rubber bands.  These shapes are called tensigrity structures first popularized by Buckminster Fuller.  The idea is that a three-dimensional structure can be built with only two kinds of elements – those in compressions (the struts) and those in tension (the rubber bands.  For example, the following image shows an octahedron built with three struts and eight rubber bands.

This simple structure is easy to assemble, and numerous other polyhedra can be built the same way.  But you might be asking where the math comes in.  This polyhedron has 8 faces, 6 vertices (corners) and 12 edges.  What relation, if any, exists between the number of faces, edges, and vertices of polyhedra?  If you can find a relationship, how can you prove it works for any polyhedron?
These questions all derive from the design and construction of a kit for making polyhedra, and the questions we ask cover quite a few of the key concepts for the Common Core math standards.
Imagine how activities like this might make math exciting for kids who currently fail to see why math can be an interesting topic!